In this subsection, we introduce the basic aspects of the Shifted Boundary Method adapted to the shallow water equations [1]}, [2]}, [3]}, [4]}. We consider a surrogate domain \(\tilde{\mathcal {D}}\)  and boundary \(\tilde{\Gamma }\)  together with the true computational domain \(\mathcal {D}\)  and its boundary \(\Gamma \) , as shown in Figure REF  and Figure REF .
We indicate by \(\mathbf {\tilde{n}}\)  the unit outward-pointing normal to the surrogate boundary \(\tilde{\Gamma }\) , which is distinct from the outward-pointing normal \(\mathbf {n}\)  to \({\Gamma }\)  as seen in Figure REF .
\(\tilde{\Gamma }\)  consists of the edges that are closest in some sense to the true boundary \(\Gamma \) , as shown in Figure REF .
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